(a) Let $k$ be a field and let $f(X)$ be an irreducible polynomial of degree $d>0$ over $k$. Prove that there exists a field $F$ containing $k$ as a subfield such that

$$f(X)=(X-\alpha) g(X)$$

where $\alpha \in F$ and $g(X) \in F[X]$. State carefully any results that you use.

(b) Let $k$ be a field and let $f(X)$ be a monic polynomial of degree $d>0$ over $k$, which is not necessarily irreducible. Prove that there exists a field $F$ containing $k$ as a subfield such that

$$f(X)=\prod_{i=1}^{d}\left(X-\alpha_{i}\right)$$

where $\alpha_{i} \in F$.

(c) Let $k=\mathbb{Z} /(p)$ for $p$ a prime, and let $f(X)=X^{p^{n}}-X$ for $n \geqslant 1$ an integer. For $F$ as in part (b), let $K$ be the set of roots of $f(X)$ in $F$. Prove that $K$ is a field.